An extremist is someone who demands that his or her views be heard while trying to silence others through violence or the threat of litigation. Extremists will almost always try to use the government to silence all criticism of their cherished beliefs. (And they almost always believe that what they are doing is for the greater good and will usher in a time of peace and prosperity.)
By that definition of extremism communist and Islamic countries immediately come to mind, but what about people in the west? Are there liberal extremists as well as right wing?
There are those who claim to be morally neutral who try and silence views they don't like--make no mistake, such people are extremists. If you believe the government should silence critics of Christianity, you are an extremist. If you believe the government should silence critics of Islam, you are an extremist.
But shouldn't certain views be silenced?
If someone is encouraging people to take the law into their own hands then the government has grounds to intervene against them. If a person is not encouraging people to take the law into their own hands then the government should not intervene (see James Allan: Free speech is truth's best hope).
Extremists will always seek positions of influence and power so they can use the government to impose their will on the rest of us.
The following is a good example of how the government has been used to make some people's lives miserable.
Extremists will almost always present their ideas in moral garb; presenting themselves as someone's saviour. (This is why we must ignore the nice sounding words, no matter how strongly they appeal to our emotions. We must ask the question, What will society be like in 50 years time if this is implemented? Often short term fixes cause long term headaches.)
Since extremists are fond of using big government as a weapon to impose their will, we must understand what governs the government. We must know what the limits of government are (particularly the federal government), so extremists can't use our own government against us.
Extremists are dangerous because virtue is not an end in itself to them. They will do what ever it takes to reach their goals. They will exacerbate tensions within society if it helps advance their agenda. If they can silence criticism of their cherished beliefs through fear, they will use terror or the threat of litigation to silence the timid. If smiling and shaking hands will enable them to get what they want, they will smile and shake hands. For them the issue is not "what's right," the issue is what is the most appropriate tactic to use to achieve their goals given the circumstances. They are "the end justifies the means" kind of people. Because they have sold their souls to the cause, everything becomes justifiable and sooner or later hell is unleashed.*